Simplicial Structures in Topology

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Preface On a dit, écrivais je (ou àpeuprès) dans une préface, que la géometrie est l’art de bien raisonner sur des figures mal faites. HENRI POINCARÉ [28] In 1954, One hundred years after Henri Poincaré’s birth, there was a special session during the International Congress of Mathematics, in the Netherlands, in honor of this great mathematician. The Russian mathematician, Pavel S. Aleksandrov, chosen to bring this about, started his speech by saying: ≪To the question of what is Poincaré’s relationship to topology, one can reply in a single sentence: he created it; but it is also possible to reply with a course of lectures in which Poincaré’s fundamental topological results would be discussed in greater or lesser detail≫ (see [3]). This is in part what we set out to do in this book. Topology is a branch of mathematics that deals with the study of the qualitative properties of figures. Johann Benedikt Listing was among the first mathematicians who dedicated themselves to studying geometry in this sense and in 1847, he published a paper [23] in which he coined the term Topology. Bernhard Riemann’s contribution to the birth of topology was also remarkable; after Riemann, Enrico Betti [5] studied manifolds through an invariant that generalizes Euler’s for convex polyhedra. Betti’s work provided Poincaré with the basis for his work in topology (called Analysis Sitûs by Poincaré). Poincaré realized the importance of his new theory and wrote some twelve papers on Analysis Sitûs; indeed, here is what he stated in the paper he wrote at the request of the Swedish mathematician Gösta Mittag-Leffler [28]: ≪Quant àmoi, toutes les voies diverses où jem’étais engagé successivement me conduisaient à l’Analysis Sitûs. J’avais besoin des données de cette science pour poursuivre mes études sur les courbes définies par les équations différentielles et pour les étendre aux équations différentielles d’ordre supérieur et en particulier à celles du problème des trois corps. J’en avais besoin pour l’étude des périodes des intégrales multiples et pour l’application de cette étude au développement de la fonction pérturbatrice. Enfin j’entrevoyais dans l’Analysis Sitûs un moyen d’aborder un problème importantedelathéorie des groupes, la recherche des groupes discrets ou des groupes finis contenus dans un groupe continu donné.≫ In his first paper on Analysis Sitûs, published in 1895 [27], Poincaré defined manifolds in spaces with dimension greater than three and introduced the basic concept of homeomorphism defined as the relation between two manifolds with the ix
x Preface same qualitative properties (see [28]). From a more up-to-date point of view, we may say that topology is the branch of mathematics concerning spaces with a certain structure (topological spaces), which are invariant under homeomorphisms; in other words, functions that are injective, surjective, and bicontinuous. The concept of homeomorphism allows us to group topological spaces into equivalence classes and consequently, to know in practical terms whether two of them are equal. This is the main purpose of topology. This book consists of six chapters. In the first one, we present the basic concepts in Topology, Group Actions and Category Theory needed for developing the remainder of the book. The part concerning Category Theory is especially important since this book has been set in categorial terminology. This chapter could also serve as a basic text for a mini-course on General Topology. In the second chapter, we study the category of simplicial complexes Csim, and two important covariant functors: the geometric realization functor from Csim to the category of topological spaces, and the homology functor from Csim to the category of graded Abelian groups. The geometric realization – |K| – of a simplicial complex K is a polyhedron, assumed to be compact throughout this book. This is the chapter closest to Poincaré’s initial paper. The Swiss mathematician Leonhard Euler was among the first ones to study one- and two-dimensional simplicial complexes; indeed, he used one-dimensional simplicial complexes and their geometric realization (graphs) in the famous problem about the seven bridges of Königsberg; later on, he also used two-dimensional simplicial complexes when he noticed that the relation v − e + f = 2, –wherev is the number of vertices, e is the number of edges, and f is the number of faces – holds for every convex polyhedron in the elementary sense (namely, every edge is common to two faces and every face leaves the entire polyhedron to one of its sides). By defining the so-called Betti Numbers for polyhedra of any dimension, Enrico Betti gave an initial generalization to the relation above; these are invariant under homeomorphims and, therefore, useful for classifying polyhedra. Based on these numbers, Poincaré developed a more complete characterization of polyhedra; in fact, in his 1895 [27] paper, Poincaré linked the Betti numbers to certain finitely generated Abelian groups associated with a polyhedron (integral homology groups of the polyhedron) and pointed out that the Betti numbers are ranks of homology groups. However, since homology groups are finitely generated Abelian groups, besides its free part (the one which gives its rank) they also have a torsion part; this too is an invariant by homeomorphisms. The combination of Betti numbers and torsion coefficients allows for a more complete analysis of polyhedra. In Chap. II, homology groups are considered strictly from the simplicial point of view; in other words, the geometric structure of polyhedra is overlooked. In order to develop the theory (which is, at this point, of algebraic nature), one needs to define concepts in Homological Algebra (a branch of Algebra that sprang partly from Algebraic Topology): among other things, we prove the important Long Exact
Page 2: Canadian Mathematical Society Soci
Page 5 and 6: Dr. Davide L. Ferrario Università
Page 8: Foreword to the English Edition Exc
Page 13 and 14: xii Preface homology groups, also w
Page 16 and 17: Contents I Fundamental Concepts ...
Page 18 and 19: Chapter I Fundamental Concepts I.1
Page 20 and 21: I.1 Topology 3 We need to show that
Page 22 and 23: I.1 Topology 5 Let X be a topologic
Page 24 and 25: I.1 Topology 7 (x, y) (x, −y) Fig
Page 26 and 27: I.1 Topology 9 that f is a homeomor
Page 28 and 29: I.1 Topology 11 We recall that an i
Page 30 and 31: I.1 Topology 13 Cx = � Xj j∈J i
Page 32 and 33: I.1 Topology 15 A Fig. I.5 Example
Page 34 and 35: I.1 Topology 17 with the topology i
Page 36 and 37: I.1 Topology 19 are closed in Y. Th
Page 38 and 39: I.1 Topology 21 is the diagonal of
Page 40 and 41: I.1 Topology 23 (I.1.38) Lemma. Let
Page 42 and 43: I.1 Topology 25 Here is an example.
Page 44 and 45: I.2 Categories 27 5. Prove that a f
Page 46 and 47: I.2 Categories 29 3. For every pair
Page 48 and 49: I.2 Categories 31 If conditions 2.
Page 50 and 51: I.2 Categories 33 Conversely, given
Page 52 and 53: I.2 Categories 35 q: B ⊔C → B
Page 54 and 55: I.2 Categories 37 (I.2.5) Lemma. Gi
Page 56 and 57: I.3 Group Actions 39 I.3 Group Acti
Page 58: I.3 Group Actions 41 S2 = {(x,y,z)
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44 II Simplicial Complexes (0, 1) (
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46 II Simplicial Complexes II.2 Abs
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48 II Simplicial Complexes II.2.1 T
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50 II Simplicial Complexes Let us r
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52 II Simplicial Complexes r = tp+(
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54 II Simplicial Complexes In a sim
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56 II Simplicial Complexes (not nec
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58 II Simplicial Complexes Let K3,3
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60 II Simplicial Complexes ordering
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62 II Simplicial Complexes are two
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64 II Simplicial Complexes 1-simple
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66 II Simplicial Complexes An infin
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68 II Simplicial Complexes 2. λn i
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70 II Simplicial Complexes Please n
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72 II Simplicial Complexes (II.3.7)
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74 II Simplicial Complexes (II.3.10
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76 II Simplicial Complexes If we do
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78 II Simplicial Complexes In the c
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80 II Simplicial Complexes Φi = {
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82 II Simplicial Complexes obtained
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84 II Simplicial Complexes (that is
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86 II Simplicial Complexes Therefor
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88 II Simplicial Complexes Exercise
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90 II Simplicial Complexes Hn(C;G)=
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92 II Simplicial Complexes Since im
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94 II Simplicial Complexes and cons
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96 II Simplicial Complexes In parti
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Chapter III Homology of Polyhedra I
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III.1 The Category of Polyhedra 101
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III.2 Homology of Polyhedra 109 Now
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III.2 Homology of Polyhedra 111 whe
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III.2 Homology of Polyhedra 113 A s
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III.2 Homology of Polyhedra 115 Fro
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III.3 Some Applications 117 where H
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III.3 Some Applications 119 we now
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III.3 Some Applications 121 has no
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III.3 Some Applications 123 Proof.
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III.4 Relative Homology 125 6. Let
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III.4 Relative Homology 127 such th
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III.5 Real Projective Spaces 129 We
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III.5 Real Projective Spaces 131 0
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III.5 Real Projective Spaces 133 No
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III.5 Real Projective Spaces 135 Le
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III.5 Real Projective Spaces 137 he
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III.5 Real Projective Spaces 139 We
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III.6 Homology of the Product of Tw
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152 IV Cohomology (IV.1.1) Theorem.
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154 IV Cohomology (IV.1.2) Remark.
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156 IV Cohomology If, starting from
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158 IV Cohomology When we apply thi
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160 IV Cohomology groups Q and R ar
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162 IV Cohomology (IV.2.2) Corollar
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164 IV Cohomology Since g and π r
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166 IV Cohomology It is easily prov
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168 IV Cohomology ∩: B p (K;Z) ×
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Chapter V Triangulable Manifolds V.
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V.1 Topological Manifolds 173 is a
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V.1 Topological Manifolds 175 |Ki|
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V.2 Closed Surfaces 177 we define C
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V.2 Closed Surfaces 179 Fig. V.6 Co
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V.2 Closed Surfaces 181 a a b a a d
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V.2 Closed Surfaces 183 where i = 1
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V.2 Closed Surfaces 185 Before proc
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V.2 Closed Surfaces 187 Simplicial
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V.3 Poincaré Duality 189 to the co
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V.3 Poincaré Duality 191 (V.3.3) T
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V.3 Poincaré Duality 193 therefore
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196 VI Homotopy Groups and define t
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198 VI Homotopy Groups is a homotop
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200 VI Homotopy Groups Proof. First
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202 VI Homotopy Groups (VI.1.10) Ex
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204 VI Homotopy Groups (VI.1.13) Th
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206 VI Homotopy Groups to the simpl
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208 VI Homotopy Groups Proof. For e
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210 VI Homotopy Groups Exercises 1.
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212 VI Homotopy Groups Proof. Since
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214 VI Homotopy Groups is precisely
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216 VI Homotopy Groups The group π
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218 VI Homotopy Groups 4. R ′ α
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220 VI Homotopy Groups and � y0 H
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222 VI Homotopy Groups Let f := F(
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224 VI Homotopy Groups be two homot
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226 VI Homotopy Groups (VI.3.16) Re
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228 VI Homotopy Groups where iA is
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230 VI Homotopy Groups f : I n g: I
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232 VI Homotopy Groups 8. Prove tha
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234 VI Homotopy Groups This derives
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236 VI Homotopy Groups Proof. The f
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References 1. J.W. Alexander - A pr
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Index H-space, 219 n-simple, 225 ab
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Index 243 Euclidean, 43 join, 47 su
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Simplicial Structures in Topology

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